SOLUTION: 17​% adults favor the use of unmanned drones by police agencies. Twelve U.S. adults are randomly selected. Find the probability that the number of U.S. adults who favor the use o

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Question 1209659: 17​% adults favor the use of unmanned drones by police agencies. Twelve U.S. adults are randomly selected. Find the probability that the number of U.S. adults who favor the use of unmanned drones by police agencies is​ (a) exactly​ three, (b) at least​ four, (c) less than eight.

Found 2 solutions by CPhill, ikleyn:
Answer by CPhill(1959)   (Show Source): You can put this solution on YOUR website!
This is a binomial probability problem. Here's how to solve it:
**1. Define the variables:**
* n = 12 (number of trials/adults selected)
* p = 0.17 (probability of success/favoring drones)
* q = 1 - p = 0.83 (probability of failure/not favoring drones)
* x = number of successes (adults favoring drones)
**2. Binomial probability formula:**
P(x) = (nCx) * p^x * q^(n-x)
where nCx is the binomial coefficient, calculated as n! / (x! * (n-x)!)
**(a) Exactly three:**
P(x = 3) = (12C3) * (0.17)^3 * (0.83)^9
P(x = 3) = (220) * (0.004913) * (0.185665)
P(x = 3) ≈ 0.201
**(b) At least four:**
This means x = 4, 5, 6, ..., 12. It's easier to calculate the complement and subtract from 1:
P(x ≥ 4) = 1 - P(x < 4)
P(x ≥ 4) = 1 - [P(x = 0) + P(x = 1) + P(x = 2) + P(x = 3)]
* P(x = 0) = (12C0) * (0.17)^0 * (0.83)^12 ≈ 0.122
* P(x = 1) = (12C1) * (0.17)^1 * (0.83)^11 ≈ 0.298
* P(x = 2) = (12C2) * (0.17)^2 * (0.83)^10 ≈ 0.312
* P(x = 3) ≈ 0.201 (calculated above)
P(x ≥ 4) = 1 - (0.122 + 0.298 + 0.312 + 0.201)
P(x ≥ 4) = 1 - 0.933
P(x ≥ 4) ≈ 0.067
**(c) Less than eight:**
This means x = 0, 1, 2, ..., 7. Again, it's easier to calculate the complement:
P(x < 8) = 1 - P(x ≥ 8)
P(x < 8) = 1 - [P(x = 8) + P(x = 9) + P(x = 10) + P(x = 11) + P(x = 12)]
Calculating each of these probabilities:
* P(x = 8) = (12C8) * (0.17)^8 * (0.83)^4 ≈ 0.0004
* P(x = 9) = (12C9) * (0.17)^9 * (0.83)^3 ≈ 0.00005
* P(x = 10), P(x = 11), and P(x = 12) will be even smaller and essentially zero for practical purposes.
P(x < 8) ≈ 1 - (0.0004 + 0.00005)
P(x < 8) ≈ 0.9995
So, P(x < 8) is very close to 1. It's highly likely that fewer than 8 adults in the sample will favor the use of drones.
Answer by ikleyn(52803)   (Show Source): You can put this solution on YOUR website!
.
17​% adults favor the use of unmanned drones by police agencies. Twelve U.S. adults are randomly selected.
Find the probability that the number of U.S. adults who favor the use of unmanned drones by police agencies is​
(a) exactly​ three,
(b) at least​ four,
(c) less than eight.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


        In the post by @CPhill,  all calculations for part  (b)  are incorrect.
        The answer for part  (b)  is incorrect, too.

        Below I placed my correct calculations. 


**(b) At least four:**

This means x = 4, 5, 6, ..., 12.  It's easier to calculate the complement and subtract from 1:

P(x ≥ 4) = 1 - P(x < 4)
P(x ≥ 4) = 1 - [P(x = 0) + P(x = 1) + P(x = 2) + P(x = 3)]

*   P(x = 0) = (12C0) * (0.17)^0 * (0.83)^12 ≈ 0.10689
*   P(x = 1) = (12C1) * (0.17)^1 * (0.83)^11 ≈ 0.26272
*   P(x = 2) = (12C2) * (0.17)^2 * (0.83)^10 ≈ 0.29595
*   P(x = 3) ≈ 0.201 (calculated above)

P(x ≥ 4) = 1 - (0.10689 + 0.26272 + 0.29595 + 0.201)
P(x ≥ 4) = 1 - 0.867
P(x ≥ 4) ≈ 0.133

Corrected.



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